In this work, we focus on the study of two major second-order statistics of fermionic Gaussian states. The first one is the variance of the von Neumann entanglement entropy eq:vN2.
The von Neumann entropy is the most fundamental entanglement measure that satisfies various interesting properties. It is an important theoretic quantity in quantum information theory and has applications in different fields, including the many-body systems and black hole radiation [1]}. Recently, the average von Neumann entropy over the fermionic Gaussian ensemble eq:nJ has been obtained as [2]}
\(\mathbb {E}\!\left[S\right]&=&\left(m+n-\frac{1}{2}\right)\psi _{0}(2m+2n)+\left(\frac{1}{4}-m\right)\psi _{0}(m+n)+\nonumber \\&&\left(\frac{1}{2}-n\right)\psi _{0}(2n)-\frac{1}{4}\psi _{0}(n)-m,\) 
